Complex-analytic quotients of algebraic G-varieties

Abstract

It is shown that any compact semistable quotient (in the sense of Heinzner and Snow) of a normal algebraic variety by a complex reductive Lie group $G$ is a good quotient. This reduces the investigation and classification of such complex-analytic quotients to the corresponding questions in the algebraic category. As a consequence of our main result, we show that every compact space in Nemirovski's class ${\mathcal{Q}}_G$ has a realisation as a good quotient, and that every complete algebraic variety in ${\mathcal{Q}}_G$ is unirational with finitely generated Cox ring and at worst rational singularities. In particular, every compact space in class ${\mathcal{Q}}_T$, where $T$ is an algebraic torus, is a toric variety.

23 pages; v2: minor corrections and additions as suggested by referee, to appear in Mathematische Annalen

References in corpus (2)

• Levi problem and semistable quotients
• Limiting behavior of trajectories of complex polynomial vector fields

Cited by in corpus (4)

• Moduli stacks of semistable sheaves and representations of Ext-quivers
• Birational geometry for d-critical loci and wall-crossing in Calabi-Yau 3-folds
• Gopakumar-Vafa invariants and wall-crossing
• Reductive quotients of klt singularities